Topology Atlas Document # qaaa-20 | Production Editor: Harriet Lord

© 2000 Copyright by Michael Gormley and T.B.M. McMaster. All rights reserved.

Question

A problem concerning embeddability

Michael Gormley and T.B.M. McMaster

Think of the family of all topological spaces on \alpha-many points [where \alpha is an infinite cardinal]. Let us call a subcollection C of these a `cross-section' if each space [on \alpha-many points] either embeds into a member of C or contains a homeomorph of a member of C.

Now put cs(\alpha) = the least cardinality of a cross-section of the spaces on \alpha-many points. What can we say about cs(\alpha)?

In particular, what is cs(\aleph0)? The Ginsburg and Sands paper [Amer. Math. Monthly 86 (1979) pp. 574-576] shows that it is at most 5, and it's fairly easy to show it is not 1; so is it 2, 3, 4 or 5?

Received by the editors: May 2, 2000